Determine whether the two functions are inverses. \[ w(x)=\frac{-2}{x-5} \text { and } z(x)=\frac{-2+5 x}{x} \]

Step 1 ：To determine if two functions are inverses of each other, we need to check if the composition of the two functions results in the identity function. The identity function is a function that always returns the same value that was used as its argument. In other words, for all x, f(x) = x.

Step 2 ：So, we need to check if \(w(z(x)) = x\) and \(z(w(x)) = x\). If both are true, then w and z are inverses of each other.

Step 3 ：Let's start by calculating \(w(z(x))\).

Step 4 ：The result of \(w(z(x))\) is x, which is the identity function. This means that the first condition for the functions to be inverses of each other is met.

Step 5 ：Now, let's calculate \(z(w(x))\) to check the second condition.

Step 6 ：The result of \(z(w(x))\) is also x, which is the identity function. This means that the second condition for the functions to be inverses of each other is also met.

Step 7 ：Since both conditions are met, we can conclude that the functions w and z are inverses of each other.

Step 8 ：\(\boxed{\text{The functions } w(x)=\frac{-2}{x-5} \text{ and } z(x)=\frac{-2+5 x}{x} \text{ are inverses of each other.}}\)